Question: Factor the following expression: $7$ $x^2$ $-38$ $x+$ $15$
Answer: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(7)}{(15)} &=& 105 \\ {a} + {b} &=& & & {-38} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $105$ and add them together. The factors that add up to ${-38}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-3}$ and ${b}$ is ${-35}$ $ \begin{eqnarray} {ab} &=& ({-3})({-35}) &=& 105 \\ {a} + {b} &=& {-3} + {-35} &=& -38 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {7}x^2 {-3}x {-35}x +{15} $ Group the terms so that there is a common factor in each group: $ ({7}x^2 {-3}x) + ({-35}x +{15}) $ Factor out the common factors: $ x(7x - 3) - 5(7x - 3) $ Notice how $(7x - 3)$ has become a common factor. Factor this out to find the answer. $(7x - 3)(x - 5)$